Calculus Examples

$lim_{x \to 4} \frac{1}{\sqrt{x} - 2} - \frac{4}{x - 4}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{\sqrt{x} - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$lim_{x \to 4} \frac{1}{\sqrt{x} - 2} \cdot \frac{x - 4}{x - 4} - \frac{4}{x - 4}$

Step 1.2

To write $- \frac{4}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x} - 2}{\sqrt{x} - 2}$ .

$lim_{x \to 4} \frac{1}{\sqrt{x} - 2} \cdot \frac{x - 4}{x - 4} - \frac{4}{x - 4} \cdot \frac{\sqrt{x} - 2}{\sqrt{x} - 2}$

Step 1.3

Write each expression with a common denominator of $(\sqrt{x} - 2) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{\sqrt{x} - 2}$ by $\frac{x - 4}{x - 4}$ .

$lim_{x \to 4} \frac{x - 4}{(\sqrt{x} - 2) (x - 4)} - \frac{4}{x - 4} \cdot \frac{\sqrt{x} - 2}{\sqrt{x} - 2}$

Step 1.3.2

Multiply $\frac{4}{x - 4}$ by $\frac{\sqrt{x} - 2}{\sqrt{x} - 2}$ .

$lim_{x \to 4} \frac{x - 4}{(\sqrt{x} - 2) (x - 4)} - \frac{4 (\sqrt{x} - 2)}{(x - 4) (\sqrt{x} - 2)}$

Step 1.3.3

Reorder the factors of $(\sqrt{x} - 2) (x - 4)$ .

$lim_{x \to 4} \frac{x - 4}{(x - 4) (\sqrt{x} - 2)} - \frac{4 (\sqrt{x} - 2)}{(x - 4) (\sqrt{x} - 2)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{x - 4 - 4 (\sqrt{x} - 2)}{(x - 4) (\sqrt{x} - 2)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 4} x - 4 - 4 (\sqrt{x} - 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - lim_{x \to 4} 4 - lim_{x \to 4} 4 (\sqrt{x} - 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.2

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 1 \cdot 4 - lim_{x \to 4} 4 (\sqrt{x} - 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 4} x - 1 \cdot 4 - 4 lim_{x \to 4} \sqrt{x} - 2}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 1 \cdot 4 - 4 (lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.5

Move the limit under the radical sign.

$\frac{lim_{x \to 4} x - 1 \cdot 4 - 4 (\sqrt{lim_{x \to 4} x} - lim_{x \to 4} 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.6

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 1 \cdot 4 - 4 (\sqrt{lim_{x \to 4} x} - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.7

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 2.1.2.7.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 1 \cdot 4 - 4 (\sqrt{lim_{x \to 4} x} - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.7.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 1 \cdot 4 - 4 (\sqrt{4} - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8

Simplify the answer.

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Step 2.1.2.8.1

Simplify each term.

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Step 2.1.2.8.1.1

Multiply $- 1$ by $4$ .

$\frac{4 - 4 - 4 (\sqrt{4} - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.1.2

Simplify each term.

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Step 2.1.2.8.1.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{4 - 4 - 4 (\sqrt{2^{2}} - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.1.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 - 4 - 4 (2 - 1 \cdot 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.1.2.3

Multiply $- 1$ by $2$ .

$\frac{4 - 4 - 4 (2 - 2)}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.1.3

Subtract $2$ from $2$ .

$\frac{4 - 4 - 4 \cdot 0}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.1.4

Multiply $- 4$ by $0$ .

$\frac{4 - 4 + 0}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.2

Subtract $4$ from $4$ .

$\frac{0 + 0}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.2.8.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 4} (x - 4) (\sqrt{x} - 2)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{lim_{x \to 4} x - 4 \cdot lim_{x \to 4} \sqrt{x} - 2}$

Step 2.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{(lim_{x \to 4} x - lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x} - 2}$

Step 2.1.3.3

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{0}{(lim_{x \to 4} x - 1 \cdot 4) \cdot lim_{x \to 4} \sqrt{x} - 2}$

Step 2.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{(lim_{x \to 4} x - 1 \cdot 4) \cdot (lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 2)}$

Step 2.1.3.5

Move the limit under the radical sign.

$\frac{0}{(lim_{x \to 4} x - 1 \cdot 4) \cdot (\sqrt{lim_{x \to 4} x} - lim_{x \to 4} 2)}$

Step 2.1.3.6

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$\frac{0}{(lim_{x \to 4} x - 1 \cdot 4) \cdot (\sqrt{lim_{x \to 4} x} - 1 \cdot 2)}$

Step 2.1.3.7

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 2.1.3.7.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{(4 - 1 \cdot 4) \cdot (\sqrt{lim_{x \to 4} x} - 1 \cdot 2)}$

Step 2.1.3.7.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{(4 - 1 \cdot 4) \cdot (\sqrt{4} - 1 \cdot 2)}$

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Multiply $- 1$ by $4$ .

$\frac{0}{(4 - 4) \cdot (\sqrt{4} - 1 \cdot 2)}$

Step 2.1.3.8.2

Subtract $4$ from $4$ .

$\frac{0}{0 \cdot (\sqrt{4} - 1 \cdot 2)}$

Step 2.1.3.8.3

Simplify each term.

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Step 2.1.3.8.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{0}{0 \cdot (\sqrt{2^{2}} - 1 \cdot 2)}$

Step 2.1.3.8.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{0 \cdot (2 - 1 \cdot 2)}$

Step 2.1.3.8.3.3

Multiply $- 1$ by $2$ .

$\frac{0}{0 \cdot (2 - 2)}$

Step 2.1.3.8.4

Subtract $2$ from $2$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.8.5

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.8.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.3.9

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 4} \frac{x - 4 - 4 (\sqrt{x} - 2)}{(x - 4) (\sqrt{x} - 2)} = lim_{x \to 4} \frac{\frac{d}{d x} [x - 4 - 4 (\sqrt{x} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [x - 4 - 4 (\sqrt{x} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.2

By the Sum Rule, the derivative of $x - 4 - 4 (\sqrt{x} - 2)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4] + \frac{d}{d x} [- 4 (\sqrt{x} - 2)]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 4] + \frac{d}{d x} [- 4 (\sqrt{x} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 4} \frac{1 + \frac{d}{d x} [- 4] + \frac{d}{d x} [- 4 (\sqrt{x} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{d}{d x} [- 4 (\sqrt{x} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5

Evaluate $\frac{d}{d x} [- 4 (\sqrt{x} - 2)]$ .

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Step 2.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{d}{d x} [- 4 (x^{\frac{1}{2}} - 2)]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 (x^{\frac{1}{2}} - 2)$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{2}} - 2]$ .

$lim_{x \to 4} \frac{1 + 0 - 4 \frac{d}{d x} [x^{\frac{1}{2}} - 2]}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.3

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2])}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 2])}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.7

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.8

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.9

Simplify the numerator.

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Step 2.3.5.9.1

Multiply $- 1$ by $2$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{1 - 2}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.9.2

Subtract $2$ from $1$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{\frac{- 1}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.10

Move the negative in front of the fraction.

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{- \frac{1}{2}} + 0)}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.11

Add $\frac{1}{2} x^{- \frac{1}{2}}$ and $0$ .

$lim_{x \to 4} \frac{1 + 0 - 4 (\frac{1}{2} x^{- \frac{1}{2}})}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.12

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 + 0 - 4 \frac{x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.13

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{- 4 x^{- \frac{1}{2}}}{2}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.14

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{- 4}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.15

Factor $2$ out of $- 4$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.16

Cancel the common factors.

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Step 2.3.5.16.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 + 0 + \frac{2 \cdot - 2}{2 (x^{\frac{1}{2}})}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.16.2

Cancel the common factor.

$lim_{x \to 4} \frac{1 + 0 + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.16.3

Rewrite the expression.

$lim_{x \to 4} \frac{1 + 0 + \frac{- 2}{x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.5.17

Move the negative in front of the fraction.

$lim_{x \to 4} \frac{1 + 0 - \frac{2}{x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.6

Add $1$ and $0$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (\sqrt{x} - 2)]}$

Step 2.3.7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{d}{d x} [(x - 4) (x^{\frac{1}{2}} - 2)]}$

Step 2.3.8

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 4$ and $g (x) = x^{\frac{1}{2}} - 2$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{d}{d x} [x^{\frac{1}{2}} - 2] + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.9

By the Sum Rule, the derivative of $x^{\frac{1}{2}} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.12

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.13

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.14

Simplify the numerator.

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Step 2.3.14.1

Multiply $- 1$ by $2$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.14.2

Subtract $2$ from $1$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.15

Move the negative in front of the fraction.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.16

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.17

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.18

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) (\frac{1}{2 x^{\frac{1}{2}}} + 0) + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.19

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 2) \frac{d}{d x} [x - 4]}$

Step 2.3.20

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])}$

Step 2.3.21

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 2) (1 + \frac{d}{d x} [- 4])}$

Step 2.3.22

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 2) (1 + 0)}$

Step 2.3.23

Add $1$ and $0$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + (x^{\frac{1}{2}} - 2) \cdot 1}$

Step 2.3.24

Multiply $x^{\frac{1}{2}} - 2$ by $1$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{(x - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25

Simplify.

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Step 2.3.25.1

Apply the distributive property.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{x \frac{1}{2 x^{\frac{1}{2}}} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2

Combine terms.

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Step 2.3.25.2.1

Combine $x$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x}{2 x^{\frac{1}{2}}} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.2

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x \cdot x^{- \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.3

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.25.2.3.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 2.3.25.2.3.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{1} x^{- \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{1 - \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.3.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.3.3

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{2 - 1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.3.4

Subtract $1$ from $2$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} - 4 \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.4

Combine $- 4$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{- 4}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.5

Factor $2$ out of $- 4$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.6

Cancel the common factors.

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Step 2.3.25.2.6.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 2}{2 (x^{\frac{1}{2}})} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.6.2

Cancel the common factor.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{2 \cdot - 2}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.6.3

Rewrite the expression.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{- 2}{x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.7

Move the negative in front of the fraction.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}} + x^{\frac{1}{2}} - 2}$

Step 2.3.25.2.8

To write $x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + x^{\frac{1}{2}} \cdot \frac{2}{2} - \frac{2}{x^{\frac{1}{2}}} - 2}$

Step 2.3.25.2.9

Combine $x^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}}}{2} + \frac{x^{\frac{1}{2}} \cdot 2}{2} - \frac{2}{x^{\frac{1}{2}}} - 2}$

Step 2.3.25.2.10

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot 2}{2} - \frac{2}{x^{\frac{1}{2}}} - 2}$

Step 2.3.25.2.11

Move $2$ to the left of $x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}} + 2 \cdot x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}} - 2}$

Step 2.3.25.2.12

Add $x^{\frac{1}{2}}$ and $2 x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{3 x^{\frac{1}{2}}}{2} - \frac{2}{x^{\frac{1}{2}}} - 2}$

Step 2.3.25.3

Reorder terms.

$lim_{x \to 4} \frac{1 - \frac{2}{x^{\frac{1}{2}}}}{\frac{3 x^{\frac{1}{2}}}{2} - 2 - \frac{2}{x^{\frac{1}{2}}}}$

Step 2.4

Convert fractional exponents to radicals.

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Step 2.4.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{\sqrt{x}}}{\frac{3 x^{\frac{1}{2}}}{2} - 2 - \frac{2}{x^{\frac{1}{2}}}}$

Step 2.4.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} - 2 - \frac{2}{x^{\frac{1}{2}}}}$

Step 2.4.3

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$lim_{x \to 4} \frac{1 - \frac{2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} - 2 - \frac{2}{\sqrt{x}}}$

Step 2.5

Combine terms.

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Step 2.5.1

Write $1$ as a fraction with a common denominator.

$lim_{x \to 4} \frac{\frac{\sqrt{x}}{\sqrt{x}} - \frac{2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} - 2 - \frac{2}{\sqrt{x}}}$

Step 2.5.2

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} - 2 - \frac{2}{\sqrt{x}}}$

Step 2.5.3

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} - 2 \cdot \frac{2}{2} - \frac{2}{\sqrt{x}}}$

Step 2.5.4

Combine $- 2$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x}}{2} + \frac{- 2 \cdot 2}{2} - \frac{2}{\sqrt{x}}}$

Step 2.5.5

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x} - 2 \cdot 2}{2} - \frac{2}{\sqrt{x}}}$

Step 2.5.6

To write $\frac{3 \sqrt{x} - 2 \cdot 2}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x} - 2 \cdot 2}{2} \cdot \frac{\sqrt{x}}{\sqrt{x}} - \frac{2}{\sqrt{x}}}$

Step 2.5.7

To write $- \frac{2}{\sqrt{x}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{3 \sqrt{x} - 2 \cdot 2}{2} \cdot \frac{\sqrt{x}}{\sqrt{x}} - \frac{2}{\sqrt{x}} \cdot \frac{2}{2}}$

Step 2.5.8

Write each expression with a common denominator of $2 \sqrt{x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.8.1

Multiply $\frac{3 \sqrt{x} - 2 \cdot 2}{2}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{(3 \sqrt{x} - 2 \cdot 2) \sqrt{x}}{2 \sqrt{x}} - \frac{2}{\sqrt{x}} \cdot \frac{2}{2}}$

Step 2.5.8.2

Multiply $\frac{2}{\sqrt{x}}$ by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{(3 \sqrt{x} - 2 \cdot 2) \sqrt{x}}{2 \sqrt{x}} - \frac{2 \cdot 2}{\sqrt{x} \cdot 2}}$

Step 2.5.8.3

Reorder the factors of $\sqrt{x} \cdot 2$ .

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{(3 \sqrt{x} - 2 \cdot 2) \sqrt{x}}{2 \sqrt{x}} - \frac{2 \cdot 2}{2 \sqrt{x}}}$

Step 2.5.9

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{\sqrt{x} - 2}{\sqrt{x}}}{\frac{(3 \sqrt{x} - 2 \cdot 2) \sqrt{x} - 2 \cdot 2}{2 \sqrt{x}}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify the limit argument.

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Step 3.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 4} \frac{\sqrt{x} - 2}{\sqrt{x}} \cdot \frac{2 \sqrt{x}}{(3 \sqrt{x} - 2 \cdot 2) \sqrt{x} - 2 \cdot 2}$

Step 3.1.2

Combine factors.

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Step 3.1.2.1

Multiply $- 2$ by $2$ .

$lim_{x \to 4} \frac{\sqrt{x} - 2}{\sqrt{x}} \cdot \frac{2 \sqrt{x}}{(3 \sqrt{x} - 4) \sqrt{x} - 2 \cdot 2}$

Step 3.1.2.2

Multiply $- 2$ by $2$ .

$lim_{x \to 4} \frac{\sqrt{x} - 2}{\sqrt{x}} \cdot \frac{2 \sqrt{x}}{(3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 3.1.2.3

Multiply $\frac{\sqrt{x} - 2}{\sqrt{x}}$ by $\frac{2 \sqrt{x}}{(3 \sqrt{x} - 4) \sqrt{x} - 4}$ .

$lim_{x \to 4} \frac{(\sqrt{x} - 2) (2 \sqrt{x})}{\sqrt{x} ((3 \sqrt{x} - 4) \sqrt{x} - 4)}$

Step 3.1.3

Cancel the common factor of $\sqrt{x}$ .

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Step 3.1.3.1

Cancel the common factor.

$lim_{x \to 4} \frac{(\sqrt{x} - 2) (2 \sqrt{x})}{\sqrt{x} ((3 \sqrt{x} - 4) \sqrt{x} - 4)}$

Step 3.1.3.2

Rewrite the expression.

$lim_{x \to 4} \frac{(\sqrt{x} - 2) \cdot (2)}{(3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 3.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 4} \frac{\sqrt{x} - 2}{(3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 4.1.1

Take the limit of the numerator and the limit of the denominator.

$2 \frac{lim_{x \to 4} \sqrt{x} - 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

Evaluate the limit.

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Step 4.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$2 \frac{lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.1.2

Move the limit under the radical sign.

$2 \frac{\sqrt{lim_{x \to 4} x} - lim_{x \to 4} 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.1.3

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$2 \frac{\sqrt{lim_{x \to 4} x} - 1 \cdot 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 \frac{\sqrt{4} - 1 \cdot 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.3

Simplify the answer.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Rewrite $4$ as $2^{2}$ .

$2 \frac{\sqrt{2^{2}} - 1 \cdot 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.3.1.2

Pull terms out from under the radical, assuming positive real numbers.

$2 \frac{2 - 1 \cdot 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.3.1.3

Multiply $- 1$ by $2$ .

$2 \frac{2 - 2}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.2.3.2

Subtract $2$ from $2$ .

$2 \frac{0}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - 4}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$2 \frac{0}{lim_{x \to 4} (3 \sqrt{x} - 4) \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$2 \frac{0}{lim_{x \to 4} 3 \sqrt{x} - 4 \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$2 \frac{0}{(lim_{x \to 4} 3 \sqrt{x} - lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 \frac{0}{(3 lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.5

Move the limit under the radical sign.

$2 \frac{0}{(3 \sqrt{lim_{x \to 4} x} - lim_{x \to 4} 4) \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.6

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$2 \frac{0}{(3 \sqrt{lim_{x \to 4} x} - 1 \cdot 4) \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 4}$

Step 4.1.3.7

Move the limit under the radical sign.

$2 \frac{0}{(3 \sqrt{lim_{x \to 4} x} - 1 \cdot 4) \cdot \sqrt{lim_{x \to 4} x} - lim_{x \to 4} 4}$

Step 4.1.3.8

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$2 \frac{0}{(3 \sqrt{lim_{x \to 4} x} - 1 \cdot 4) \cdot \sqrt{lim_{x \to 4} x} - 1 \cdot 4}$

Step 4.1.3.9

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 4.1.3.9.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 \frac{0}{(3 \sqrt{4} - 1 \cdot 4) \cdot \sqrt{lim_{x \to 4} x} - 1 \cdot 4}$

Step 4.1.3.9.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 \frac{0}{(3 \sqrt{4} - 1 \cdot 4) \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10

Simplify the answer.

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Step 4.1.3.10.1

Simplify each term.

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Step 4.1.3.10.1.1

Simplify each term.

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Step 4.1.3.10.1.1.1

Rewrite $4$ as $2^{2}$ .

$2 \frac{0}{(3 \sqrt{2^{2}} - 1 \cdot 4) \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10.1.1.2

Pull terms out from under the radical, assuming positive real numbers.

$2 \frac{0}{(3 \cdot 2 - 1 \cdot 4) \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10.1.1.3

Multiply $3$ by $2$ .

$2 \frac{0}{(6 - 1 \cdot 4) \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10.1.1.4

Multiply $- 1$ by $4$ .

$2 \frac{0}{(6 - 4) \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10.1.2

Subtract $4$ from $6$ .

$2 \frac{0}{2 \cdot \sqrt{4} - 1 \cdot 4}$

Step 4.1.3.10.1.3

Rewrite $4$ as $2^{2}$ .

$2 \frac{0}{2 \cdot \sqrt{2^{2}} - 1 \cdot 4}$

Step 4.1.3.10.1.4

Pull terms out from under the radical, assuming positive real numbers.

$2 \frac{0}{2 \cdot 2 - 1 \cdot 4}$

Step 4.1.3.10.1.5

Multiply $2$ by $2$ .

$2 \frac{0}{4 - 1 \cdot 4}$

Step 4.1.3.10.1.6

Multiply $- 1$ by $4$ .

$2 \frac{0}{4 - 4}$

Step 4.1.3.10.2

Subtract $4$ from $4$ .

$2 (\frac{0}{0})$

Step 4.1.3.10.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 4.1.3.11

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 4.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 (\frac{0}{0})$

Step 4.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 4} \frac{\sqrt{x} - 2}{(3 \sqrt{x} - 4) \sqrt{x} - 4} = lim_{x \to 4} \frac{\frac{d}{d x} [\sqrt{x} - 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$2 lim_{x \to 4} \frac{\frac{d}{d x} [\sqrt{x} - 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.2

By the Sum Rule, the derivative of $\sqrt{x} - 2$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 2]$ .

$2 lim_{x \to 4} \frac{\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3

Evaluate $\frac{d}{d x} [\sqrt{x}]$ .

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Step 4.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.4

Combine $- 1$ and $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.5

Combine the numerators over the common denominator.

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.6

Simplify the numerator.

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Step 4.3.3.6.1

Multiply $- 1$ by $2$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.6.2

Subtract $2$ from $1$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.3.7

Move the negative in front of the fraction.

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- 2]}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} x^{- \frac{1}{2}} + 0}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.5

Simplify.

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Step 4.3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.5.2

Combine terms.

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Step 4.3.5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}} + 0}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.5.2.2

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x} - 4]}$

Step 4.3.6

By the Sum Rule, the derivative of $(3 \sqrt{x} - 4) \sqrt{x} - 4$ with respect to $x$ is $\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x}] + \frac{d}{d x} [- 4]$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x}] + \frac{d}{d x} [- 4]}$

Step 4.3.7

Evaluate $\frac{d}{d x} [(3 \sqrt{x} - 4) \sqrt{x}]$ .

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Step 4.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 x^{\frac{1}{2}} - 4) \sqrt{x}] + \frac{d}{d x} [- 4]}$

Step 4.3.7.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{d}{d x} [(3 x^{\frac{1}{2}} - 4) x^{\frac{1}{2}}] + \frac{d}{d x} [- 4]}$

Step 4.3.7.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3 x^{\frac{1}{2}} - 4$ and $g (x) = x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{d}{d x} [x^{\frac{1}{2}}] + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{\frac{1}{2}} - 4] + \frac{d}{d x} [- 4]}$

Step 4.3.7.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} \frac{d}{d x} [3 x^{\frac{1}{2}} - 4] + \frac{d}{d x} [- 4]}$

Step 4.3.7.5

By the Sum Rule, the derivative of $3 x^{\frac{1}{2}} - 4$ with respect to $x$ is $\frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [- 4]$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (\frac{d}{d x} [3 x^{\frac{1}{2}}] + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- 4]}$

Step 4.3.7.6

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{1}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- 4]}$

Step 4.3.7.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- 4]}$

Step 4.3.7.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.10

Combine $- 1$ and $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.11

Combine the numerators over the common denominator.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.12

Simplify the numerator.

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Step 4.3.7.12.1

Multiply $- 1$ by $2$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{1 - 2}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.12.2

Subtract $2$ from $1$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{\frac{- 1}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.13

Move the negative in front of the fraction.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2} x^{- \frac{1}{2}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.14

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{x^{- \frac{1}{2}}}{2} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.15

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.16

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.17

Combine $- 1$ and $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.18

Combine the numerators over the common denominator.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1 - 1 \cdot 2}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.19

Simplify the numerator.

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Step 4.3.7.19.1

Multiply $- 1$ by $2$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{1 - 2}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.19.2

Subtract $2$ from $1$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{\frac{- 1}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.20

Move the negative in front of the fraction.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 (\frac{1}{2}) x^{- \frac{1}{2}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.21

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (3 \frac{x^{- \frac{1}{2}}}{2} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.22

Combine $3$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (\frac{3 x^{- \frac{1}{2}}}{2} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.23

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} (\frac{3}{2 x^{\frac{1}{2}}} + 0) + \frac{d}{d x} [- 4]}$

Step 4.3.7.24

Add $\frac{3}{2 x^{\frac{1}{2}}}$ and $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{2}} \frac{3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 4.3.7.25

Combine $x^{\frac{1}{2}}$ and $\frac{3}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}} \cdot 3}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 4.3.7.26

Move $3$ to the left of $x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3 \cdot x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 4.3.7.27

Cancel the common factor.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 4]}$

Step 4.3.7.28

Rewrite the expression.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} + \frac{3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.29

To write $(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{2}{2} + \frac{3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.30

Combine $(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}}$ and $\frac{2}{2}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} \cdot 2}{2} + \frac{3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.31

Combine the numerators over the common denominator.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{1}{2 x^{\frac{1}{2}}} \cdot 2 + 3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.32

Combine $2$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{2}{2 x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.33

Cancel the common factor.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{2}{2 x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.7.34

Rewrite the expression.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{1}{x^{\frac{1}{2}}} + 3}{2} + \frac{d}{d x} [- 4]}$

Step 4.3.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{(3 x^{\frac{1}{2}} - 4) \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9

Simplify.

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Step 4.3.9.1

Apply the distributive property.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{2}}} - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2

Combine terms.

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Step 4.3.9.2.1

Combine $3$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{x^{\frac{1}{2}} \frac{3}{x^{\frac{1}{2}}} - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.2

Combine $x^{\frac{1}{2}}$ and $\frac{3}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{x^{\frac{1}{2}} \cdot 3}{x^{\frac{1}{2}}} - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.3

Move $3$ to the left of $x^{\frac{1}{2}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{3 \cdot x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.4

Cancel the common factor.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{\frac{3 x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.5

Divide $3$ by $1$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - 4 \frac{1}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.6

Combine $- 4$ and $\frac{1}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 + \frac{- 4}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.7

Move the negative in front of the fraction.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - \frac{4}{x^{\frac{1}{2}}} + 3}{2} + 0}$

Step 4.3.9.2.8

Add $3$ and $3$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{6 - \frac{4}{x^{\frac{1}{2}}}}{2} + 0}$

Step 4.3.9.2.9

Factor $2$ out of $6$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 \cdot 3 - \frac{4}{x^{\frac{1}{2}}}}{2} + 0}$

Step 4.3.9.2.10

Factor $2$ out of $- \frac{4}{x^{\frac{1}{2}}}$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 \cdot 3 + 2 \cdot - 1 \frac{2}{x^{\frac{1}{2}}}}{2} + 0}$

Step 4.3.9.2.11

Factor $2$ out of $2 (3) + 2 (- \frac{2}{x^{\frac{1}{2}}})$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{2}{x^{\frac{1}{2}}})}{2} + 0}$

Step 4.3.9.2.12

Cancel the common factors.

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Step 4.3.9.2.12.1

Factor $2$ out of $2$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{2}{x^{\frac{1}{2}}})}{2 (1)} + 0}$

Step 4.3.9.2.12.2

Cancel the common factor.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{2 (3 - \frac{2}{x^{\frac{1}{2}}})}{2 \cdot 1} + 0}$

Step 4.3.9.2.12.3

Rewrite the expression.

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{\frac{3 - \frac{2}{x^{\frac{1}{2}}}}{1} + 0}$

Step 4.3.9.2.12.4

Divide $3 - \frac{2}{x^{\frac{1}{2}}}$ by $1$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{3 - \frac{2}{x^{\frac{1}{2}}} + 0}$

Step 4.3.9.2.13

Add $3 - \frac{2}{x^{\frac{1}{2}}}$ and $0$ .

$2 lim_{x \to 4} \frac{\frac{1}{2 x^{\frac{1}{2}}}}{3 - \frac{2}{x^{\frac{1}{2}}}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$2 lim_{x \to 4} \frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{1}{3 - \frac{2}{x^{\frac{1}{2}}}}$

Step 4.5

Convert fractional exponents to radicals.

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Step 4.5.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$2 lim_{x \to 4} \frac{1}{2 \sqrt{x}} \cdot \frac{1}{3 - \frac{2}{x^{\frac{1}{2}}}}$

Step 4.5.2

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$2 lim_{x \to 4} \frac{1}{2 \sqrt{x}} \cdot \frac{1}{3 - \frac{2}{\sqrt{x}}}$

Step 4.6

Multiply $\frac{1}{2 \sqrt{x}}$ by $\frac{1}{3 - \frac{2}{\sqrt{x}}}$ .

$2 lim_{x \to 4} \frac{1}{2 \sqrt{x} (3 - \frac{2}{\sqrt{x}})}$

Step 4.7

Combine terms.

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Step 4.7.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$2 lim_{x \to 4} \frac{1}{2 \sqrt{x} (\frac{3 \sqrt{x}}{\sqrt{x}} - \frac{2}{\sqrt{x}})}$

Step 4.7.2

Combine the numerators over the common denominator.

$2 lim_{x \to 4} \frac{1}{2 \sqrt{x} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$2 (\frac{1}{2}) lim_{x \to 4} \frac{1}{\sqrt{x} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{lim_{x \to 4} 1}{lim_{x \to 4} \sqrt{x} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5.3

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{1}{lim_{x \to 4} \sqrt{x} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5.4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{1}{lim_{x \to 4} \sqrt{x} \cdot lim_{x \to 4} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5.5

Move the limit under the radical sign.

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot lim_{x \to 4} \frac{3 \sqrt{x} - 2}{\sqrt{x}}}$

Step 5.6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{lim_{x \to 4} 3 \sqrt{x} - 2}{lim_{x \to 4} \sqrt{x}}}$

Step 5.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{lim_{x \to 4} 3 \sqrt{x} - lim_{x \to 4} 2}{lim_{x \to 4} \sqrt{x}}}$

Step 5.8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{3 lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 2}{lim_{x \to 4} \sqrt{x}}}$

Step 5.9

Move the limit under the radical sign.

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{3 \sqrt{lim_{x \to 4} x} - lim_{x \to 4} 2}{lim_{x \to 4} \sqrt{x}}}$

Step 5.10

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{3 \sqrt{lim_{x \to 4} x} - 1 \cdot 2}{lim_{x \to 4} \sqrt{x}}}$

Step 5.11

Move the limit under the radical sign.

$2 (\frac{1}{2}) \frac{1}{\sqrt{lim_{x \to 4} x} \cdot \frac{3 \sqrt{lim_{x \to 4} x} - 1 \cdot 2}{\sqrt{lim_{x \to 4} x}}}$

Step 6

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{lim_{x \to 4} x} - 1 \cdot 2}{\sqrt{lim_{x \to 4} x}}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{lim_{x \to 4} x}}}$

Step 6.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$2 (\frac{1}{2}) \frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{4}}}$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Cancel the common factor.

$2 (\frac{1}{2}) \frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{4}}}$

Step 7.1.2

Rewrite the expression.

$1 \frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{4}}}$

Step 7.2

Multiply $\frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{4}}}$ by $1$ .

$\frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{4} - 1 \cdot 2}{\sqrt{4}}}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{\sqrt{4} \cdot \frac{3 \sqrt{2^{2}} - 1 \cdot 2}{\sqrt{4}}}$

Step 7.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{\sqrt{4} \cdot \frac{3 \cdot 2 - 1 \cdot 2}{\sqrt{4}}}$

Step 7.3.3

Multiply $3$ by $2$ .

$\frac{1}{\sqrt{4} \cdot \frac{6 - 1 \cdot 2}{\sqrt{4}}}$

Step 7.3.4

Multiply $- 1$ by $2$ .

$\frac{1}{\sqrt{4} \cdot \frac{6 - 2}{\sqrt{4}}}$

Step 7.3.5

Subtract $2$ from $6$ .

$\frac{1}{\sqrt{4} \cdot \frac{4}{\sqrt{4}}}$

Step 7.4

Simplify the denominator.

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Step 7.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{\sqrt{4} \cdot \frac{4}{\sqrt{2^{2}}}}$

Step 7.4.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{\sqrt{4} \cdot \frac{4}{2}}$

Step 7.5

Combine $\sqrt{4}$ and $\frac{4}{2}$ .

$\frac{1}{\frac{\sqrt{4} \cdot 4}{2}}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{\frac{\sqrt{2^{2}} \cdot 4}{2}}$

Step 7.6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{\frac{2 \cdot 4}{2}}$

Step 7.7

Multiply $2$ by $4$ .

$\frac{1}{\frac{8}{2}}$

Step 7.8

Divide $8$ by $2$ .

$\frac{1}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$